Isogonal Conjugate
The isogonal conjugate 
of a point 
in the plane of the Triangle 
is constructed byreflecting the lines 
, 
, and 
about the Angle Bisectors at 
, 
, and 
. Thethree reflected lines Concur at the isogonal conjugate. The Trilinear Coordinates of theisogonal conjugate of the point with coordinates
are
Isogonal conjugation maps the interior of a Triangle onto itself. This mapping transforms lines onto ConicSections that Circumscribe the Triangle. The type of Conic Sectionis determined by whether the line 
meets the Circumcircle 
,
- 1. If does not intersect , the isogonal transform is an Ellipse;
- 2. If is tangent to , the transform is a Parabola;
- 3. If cuts , the transform is a Hyperbola, which is a Rectangular Hyperbola if the line passesthrough the Circumcenter
The isogonal conjugate of a point on the Circumcircle is a Point at Infinity (and conversely). The sides ofthe Pedal Triangle of a point are Perpendicular to the connectors of the correspondingVertices with the isogonal conjugate. The isogonal conjugate of a set of points is theLocus of their isogonal conjugate points.
The product of Isotomic and isogonal conjugation is a Collineation whichtransforms the sides of a Triangle to themselves (Vandeghen 1965).
See also Antipedal Triangle, Collineation, Isogonal Line,Isotomic Conjugate Point, Line at Infinity, Symmedian Line
References
Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 153-158, 1929. Vandeghen, A. ``Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle.'' Amer. Math. Monthly 72, 1091-1094, 1965.
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